Multiplexity and multireciprocity in directed multiplexes

Gemmetto, Valerio and Squartini, Tiziano and Picciolo, Francesco and Ruzzenenti, Franco and Garlaschelli, Diego Multiplexity and multireciprocity in directed multiplexes. Physical Review E. ISSN 1539-3755 (2015)

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Abstract

In recent years, the study of multi-layer networks has received much attention. Here, we provide new measures of dependency between directed links across different layers of multiplex networks. We show that this operation requires more than a straightforward extension of the corresponding multiplexity measures that have been developed for undirected multiplexes. In particular, one should take into account the effects of reciprocity, i.e. the tendency of pairs of vertices to establish mutual connections. We extend this quantity to multiplexes and introduce the notion of multireciprocity, defined as the tendency of links in one layer to be reciprocated by links in a different layer. While ordinary reciprocity reduces to a scalar quantity, multireciprocity requires a square matrix generated by all the possible pairs of layers. We introduce multireciprocity metrics valid for both binary and weighted networks and then measure these quantities on the World Trade Multiplex (WTM), representing the import-export relationships between world countries in different products. We show that several pairs of layers exhibit strong multiplexity, an effect which can however be largely encoded into the degree or strength sequences of individual layers. We also find that most pairs of commodities are characterised by positive multireciprocity, and that such values are significantly lower than the usual reciprocity measured on the aggregated network. We finally identify robust empirical patterns that allow us to use the multireciprocity matrix to retrieve the two-layer reciprocated degree (strength) of a node from the ordinary in-degree (in-strength) in a single layer and to reconstruct joint multi-layer connection probabilities from marginal ones, hence bridging the gap between single-layer properties and truly multiplex information.

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